3.2.41 \(\int \frac {\sqrt {b \sec (c+d x)}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) [141]

Optimal. Leaf size=98 \[ \frac {3 x \sqrt {b \sec (c+d x)}}{8 \sqrt {\sec (c+d x)}}+\frac {\sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {3 \sqrt {b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)} \]

[Out]

1/4*sin(d*x+c)*(b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(7/2)+3/8*sin(d*x+c)*(b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(3/2)+
3/8*x*(b*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 2715, 8} \begin {gather*} \frac {3 x \sqrt {b \sec (c+d x)}}{8 \sqrt {\sec (c+d x)}}+\frac {3 \sin (c+d x) \sqrt {b \sec (c+d x)}}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\sin (c+d x) \sqrt {b \sec (c+d x)}}{4 d \sec ^{\frac {7}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(9/2),x]

[Out]

(3*x*Sqrt[b*Sec[c + d*x]])/(8*Sqrt[Sec[c + d*x]]) + (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(7/2
)) + (3*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 17

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[a^(m + 1/2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v])
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps

\begin {align*} \int \frac {\sqrt {b \sec (c+d x)}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {\sqrt {b \sec (c+d x)} \int \cos ^4(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=\frac {\sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (3 \sqrt {b \sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt {\sec (c+d x)}}\\ &=\frac {\sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {3 \sqrt {b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (3 \sqrt {b \sec (c+d x)}\right ) \int 1 \, dx}{8 \sqrt {\sec (c+d x)}}\\ &=\frac {3 x \sqrt {b \sec (c+d x)}}{8 \sqrt {\sec (c+d x)}}+\frac {\sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {3 \sqrt {b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 55, normalized size = 0.56 \begin {gather*} \frac {\sqrt {b \sec (c+d x)} (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{32 d \sqrt {\sec (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(9/2),x]

[Out]

(Sqrt[b*Sec[c + d*x]]*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(32*d*Sqrt[Sec[c + d*x]])

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Maple [A]
time = 36.39, size = 74, normalized size = 0.76

method	result	size
default	\(\frac {\left (2 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 d x +3 c \right ) \sqrt {\frac {b}{\cos \left (d x +c \right )}}}{8 d \cos \left (d x +c \right )^{4} \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}}}\)	\(74\)
risch	\(\frac {3 \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, x}{8 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, {\mathrm e}^{2 i \left (d x +c \right )}}{8 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}+\frac {i \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, {\mathrm e}^{-2 i \left (d x +c \right )}}{8 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}+\frac {\sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \sin \left (4 d x +4 c \right )}{32 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/8/d*(2*cos(d*x+c)^3*sin(d*x+c)+3*sin(d*x+c)*cos(d*x+c)+3*d*x+3*c)*(b/cos(d*x+c))^(1/2)/cos(d*x+c)^4/(1/cos(d
*x+c))^(9/2)

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Maxima [A]
time = 0.63, size = 49, normalized size = 0.50 \begin {gather*} \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} \sqrt {b}}{32 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/32*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))))*sqrt(b)/d

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Fricas [A]
time = 4.05, size = 202, normalized size = 2.06 \begin {gather*} \left [\frac {\frac {2 \, {\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}} + 3 \, \sqrt {-b} \log \left (-2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{16 \, d}, \frac {\frac {{\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}} + 3 \, \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{8 \, d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

[1/16*(2*(2*cos(d*x + c)^4 + 3*cos(d*x + c)^2)*sqrt(b/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)) + 3*sqrt(-
b)*log(-2*sqrt(-b)*sqrt(b/cos(d*x + c))*cos(d*x + c)^(3/2)*sin(d*x + c) + 2*b*cos(d*x + c)^2 - b))/d, 1/8*((2*
cos(d*x + c)^4 + 3*cos(d*x + c)^2)*sqrt(b/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)) + 3*sqrt(b)*arctan(sqr
t(b/cos(d*x + c))*sin(d*x + c)/(sqrt(b)*sqrt(cos(d*x + c)))))/d]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))**(1/2)/sec(d*x+c)**(9/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(d*x + c))/sec(d*x + c)^(9/2), x)

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Mupad [B]
time = 0.71, size = 52, normalized size = 0.53 \begin {gather*} \frac {\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\left (8\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (4\,c+4\,d\,x\right )+12\,d\,x\right )}{32\,d\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/cos(c + d*x))^(1/2)/(1/cos(c + d*x))^(9/2),x)

[Out]

((b/cos(c + d*x))^(1/2)*(8*sin(2*c + 2*d*x) + sin(4*c + 4*d*x) + 12*d*x))/(32*d*(1/cos(c + d*x))^(1/2))

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